ABSTRACT

Introduction

Random coefficient models

Motivating Examples

Random coefficient vs. Basic model

In the absence of a Bayesian hierarchical model, there are two approaches for this problem:

Complete pooling and No pooling

Partial pooled model/Random coefficient

Research Objectives

Methodology

Dynamic Modeling of traffic conditions

\[P_{ij}=(P_{t+1}=j|P_{t}=i)\]

\[P_{ij} = \left[\begin{array} {rr} P_{ff} & P_{fc} \\ P_{cf} & P_{cc} \\ \end{array}\right] \]

\[\sum_{j=1}^{2} P_{ij}=1\]

\[P_{ij} = \left[\begin{array} {rr} 1-P_{fc} & P_{fc} \\ 1-P_{cc} & P_{cc} \\ \end{array}\right] \]

Issues Addressed

Time-varying effect

Lane and day of the week varying-effect

Modeling Approach

Model 1

\[ Y_{ij} = Bernoulli(\pi_{ij})\] \[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \beta X + \epsilon_{k}\]

\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]

\[ \sigma_{1} \sim unif(0, 100)\] \[ \beta \sim N(\mu=0, \sigma = 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]

Model 2

\[ Y_{ij} = Bernoulli(\pi_{ij})\]
\[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \epsilon_{k}\]

\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]

Model Comparison

\[ WAIC = -2*lppd + 2*pWAIC\]
where,

          lppd is the log point-wise posterior density
          pWAIC is the effective number of parameters

Results

Random coefficient Lane lateral location
Median lane 46.72
Inner-left lane 51.42
Inner-right lane 50.93
Shoulder lane 57.26
Overall 52.23

Results

Results

Results

Results

  • No significant difference in model fit between CIRS and RC for the SCR data

  • RIS was the best in fitting Breakdown data

RC was selected for modeling Breakdown and SCR data

Results

  • Model Comparison Results

Results

  • Remaining in the congested regime